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Encyclopedia > John Wallis
John Wallis

John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is also credited with introducing the symbol ∞ for infinity. Download high resolution version (843x867, 230 KB) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... Download high resolution version (843x867, 230 KB) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... November 22 is the 326th day (327th on leap years) of the year in the Gregorian calendar. ... Events Dirk Hartog lands on an island off the Western Australian coast Pocahontas arrives in England War between Venice and Austria Collegium Musicum founded in Prague Nicolaus Copernicus De revolutionibus is placed on the Index of Forbidden Books by the Roman Catholic Church Births May 18 - Johann Jakob Froberger, German... October 28 is the 301st day of the year (302nd in leap years) in the Gregorian Calendar, with 64 days remaining. ... Events February 2 - Earthquake in Aquila, Italy February 4 - In Japan, the 47 samurai commit seppuku (ritual suicide) February 14 - Earthquake in Norcia, Italy May 27 - Founding of St Petersburg in Russia May 26 - Portugal joins Grand Alliance July 29-31 - Daniel Defoe is placed in a pillory for the... Royal motto: Dieu et mon droit (French: God and my right) Englands location within the UK Official language English de facto Capital London de facto Largest city London Area  - Total Ranked 1st UK 130,395 km² Population  - Total (2001)  - Density Ranked 1st UK 49,138,831 377/km² Religion... A mathematician is a person whose area of study and research is mathematics. ... For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... Events January 21 - Abel Tasman discovers Tonga February 6 - Abel Tasman discovers the Fiji islands. ... Events Louis XIV of France passed the Code Noir, allowing the full use of slaves in the French colonies. ... Pre-19th century Leone Battista Alberti, polymath/universal genius, inventor of polyalphabetic substitution (see frequency analysis for the significance of this -- missed by most for a long time and dumbed down in the Vigenère cipher), and what may have been the first mechanical encryption aid. ... The Parliament of the United Kingdom of Great Britain and Northern Ireland is the supreme legislative institution in the United Kingdom and British overseas territories (it alone has parliamentary sovereignty). ... Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ...


Life

John Wallis was born at Ashford, Kent, the third of five children of Reverend John Wallis and Joanna Chapman. He was initially educated at a local Ashford school, but moved to James Movat's school at Tenterden in 1625 following an outbreak of plague. Wallis was first exposed to mathematics in 1631, at Martin Holbeach's school in Felsted; he enjoyed the subject, but his study was erratic, since "mathematics, at that time with us, were scarce looked on as academical studies, but rather mechanical" (Scriba 1970). Location within the British Isles. ... Kent is a county in England, south-east of London. ... Tenterden is a small town in the Weald of Kent, England. ... Events March 27 - Prince Charles Stuart becomes King Charles I of England, Scotland and Ireland. ... Plague redirects here. ... Events February 5 - Roger Williams emigrates to Boston. ... Felsted is a village of Essex in England, located near Braintree, Great Dunmow, and Chelmsford. ...


As it was intended that he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge. While there he kept an act on the doctrine of the circulation of the blood; that was said to have been the first occasion in Europe on which this theory was publicly maintained in a disputation. His interests, however, centred on mathematics. He received his Bachelor of Arts degree in 1637, and a Master's in 1640, afterwards entering the priesthood. Wallis was elected to a fellowship at Queens' College, Cambridge in 1644, which he however had to resign following his marriage on 14 March 1645 to Susanna Glyde. Full name Emmanuel College Motto - Named after Immanuel Previous names - Established 1584 Sister College Exeter College Master The Lord Wilson of Dinton Location Regent Street Undergraduates 494 Graduates 98 Homepage Boatclub Emmanuel front court and the Wren chapel Emmanuel College is a constituent college of the University of Cambridge, founded... Full name The Queens College of Saint Margaret and Saint Bernard, commonly called Queens College, in the University of Cambridge Motto Floreat Domus May this House Flourish Named after - Previous names - Established 1448 Sister College Pembroke College President Lord Eatwell JCR President {{{JCR President}}} Location Silver Street Undergraduates 490... Events February to August - Explorer Abel Tasmans second expedition for the Dutch East India Company maps the north coast of Australia. ... March 14 is the 73rd day of the year in the Gregorian Calendar (74th in Leap years) with 292 days remaining in the year. ... Events January 10 - Archbishop Laud executed on Tower Hill. ...


Throughout this time, Wallis had been close to the Puritan party, to whom he rendered great assistance in deciphering the royalist dispatches. The quality of cryptography at that time was mixed; despite the individual successes of mathematicians such as François Viète, the principles underlying cipher design and analysis were very poorly understood. Most ciphers were ad-hoc methods relying on a secret algorithm, as opposed to systems based on a variable key. Wallis realised that the latter were far more secure — even describing them as "unbreakable", though he was not confident enough in this assertion to encourage revealing cryptographic algorithms. He was also concerned about the use of ciphers by foreign powers; refusing, for example, Gottfried Leibniz's request of 1697 to teach Hanoverian students about cryptography. The Puritans were members of a group of radical Protestants which developed in England after the Reformation. ... François Viète. ... Flowcharts are often used to represent algorithms. ... A key is a piece of information that controls the operation of a cryptography algorithm. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (also Leibnitz) (Leipzig July 1, 1646 – November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Events September 20 - The Treaty of Ryswick December 2 – St Pauls Cathedral opened in London Peter the Great travels in Europe officially incognito as artilleryman Pjotr Mikhailov Use of palanquins increases in Europe Christopher Polhem starts Swedens first technical school. ... Hanover (German Hannover) is a historical territory in todays Germany. ...


Returning to London — he had been made chaplain at St Gabriel, Fenchurch Street, in 1643 — Wallis joined the group of scientists that was later to evolve into the Royal Society. He was finally able to indulge his mathematical interests, mastering William Oughtred's Clavis Mathematicae in a few weeks in 1647. He soon began to write his own treatises, dealing with a wide range of topics: throughout his life, Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. The Clock Tower of the Palace of Westminster which contains Big Ben London is the capital city of the United Kingdom and of England. ... A chaplain is a priest or military unit, a private chapel, a ship, a prison, a hospital, a parliament and so on. ... The Royal Society of London is claimed to be the oldest learned society still in existence. ... William Oughtred (March 5, 1575 – June 30, 1660) was an English mathematician. ... Events March 14 - Thirty Years War: Bavaria, Cologne, France and Sweden sign the Truce of Ulm. ... Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. ... For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Thales (circa 624-547 BC) dealing with spatial relationships. ... In mathematics, a series is a sum of a sequence of terms. ...


John Wallis joined the moderate Presbyterians in signing the remonstrance against the execution of Charles I, by which he incurred the lasting hostility of the Independents. In spite of their opposition he was appointed in 1649 to be the Savilian Chair of Geometry at Oxford University, where he lived until his death on October 28, 1703. Besides his mathematical works he wrote on theology, logic, and philosophy, and he was the first to devise a system for teaching deaf-mutes. Charles I (19 November 1600–30 January 1649) was King of England, Scotland and Ireland from 27 March 1625, until his death. ... Events January 30 - King Charles I of England, Scotland and Ireland is beheaded. ... The University of Oxford, situated in the city of Oxford in England, is the oldest university in the English-speaking world. ... Theology is literally reasonable discourse concerning God (Greek θεος, theos, God, + λογος, logos, word or reason). By extension, it also refers to the study of other religious topics. ... Logic (from ancient Greek λόγος (logos), originally meaning the word, or what is spoken, but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy amongst philosophers (see below). ... The term Philosophy derives from a combination of the Greek words philos meaning love and sophia meaning wisdom. ...


Mathematics

In 1655, Wallis published a treatise on conic sections in which they were defined analytically. This was the earliest book in which these curves are considered and defined as curves of the second degree. It helped to remove some of the perceived difficulty and obscurity of René Descartes' work on analytic geometry. René Descartes René Descartes (IPA: , March 31, 1596 – February 11, 1650), also known as Cartesius, worked as a philosopher and mathematician. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ...


Arithmetica Infinitorum, the most important of Wallis's works, was published in 1656. In this treatise the methods of analysis of Descartes and Cavalieri were systematised and extended, but some ideals were open to criticism. He begins, after a short tract on conic sections, by developing the standard notation for powers, extending them from positive integers to rational numbers:

  • x0 = 1, x − 1 = 1 / x, x − 2 = 1 / x2, etc.
  • x1 / 2 = square root of x, x2 / 3 = cube root of x2, etc.
  • x1 / n = nth root of x.
  • xp / q = qth root of xp.

Leaving the numerous algebraic applications of this discovery, he next proceeds to find, by the method of indivisibles, the area enclosed between the curve y = xm, the axis of x, and any ordinate x = h, and he proves that the ratio of this area to that of the parallelogram on the same base and of the same altitude is 1/(m + 1). He apparently assumed that the same result would be true also for the curve y = axm, where a is any constant, and m any number positive or negative; but he only discusses the case of the parabola in which m = 2, and that of the hyperbola in which m = −1. In the latter case, his interpretation of the result is incorrect. He then shows that similar results might be written down for any curve of the form

and hence that, if the ordinate y of a curve can be expanded in powers of x, its area can be determined: thus he says that if the equation of the curve is y = x0 + x1 + x2 + ..., its area would be x + x2/2 + x3/3 + ... He then applies this to the quadrature of the curves y = (xx2)0, y = (xx2)1, y = (xx2)2, etc., taken between the limits x = 0 and x = 1. He shows that the areas are respectively 1, 1/6, 1/30, 1/140, etc. He next considers curves of the form y = x1/m and establishes the theorem that the area bounded by this curve and the lines x = 0 and x = 1 is equal to the area of the rectangle on the same base and of the same altitude as m : m + 1. This is equivalent to computing

He illustrates this by the parabola, in which case m = 2. He states, but does not prove, the corresponding result for a curve of the form y = xp/q.


Wallis showed considerable ingenuity in reducing the equations of curves to the forms given above, but, as he was unacquainted with the binomial theorem, he could not effect the quadrature of the circle, whose equation is y =, since he was unable to expand this in powers of x. He laid down, however, the principle of interpolation. Thus, as the ordinate of the circle y = is the geometrical mean between the ordinates of the curves y = and y =, it might be suppose that, as an approximation, the area of the semicircle dx which is might be taken as the geometrical mean between the values of In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

 dx and dx 

that is, 1 and ; this is equivalent to taking or 3.26... as the value of . But, Wallis argued, we have in fact a series 1,,,,... and therefore the term interpolated between 1 and ought to be chosen so as to obey the law of this series. This, by an elaborate method, which I need not describe in detail, leads to a value for the interpolated term which is equivalent to taking [ <-- This needs a lot of work! ]


In this work also the formation and properties of continued fractions are discussed, the subject having been brought into prominence by Brouncker's use of these fractions. In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Lord William Brouncker (born 1620 in Castlelyons, County Cork, Ireland and died on 5 April 1684 in Westminster, London, UK) was an English mathematician. ...


A few years later, in 1659, Wallis published a tract containing the solution of the problems on the cycloid which had been proposed by Blaise Pascal. In this he incidentally explained how the principles laid down in his Arithmetica Infinitorum could be used for the rectification of algebraic curves; and gave a solution of the problem to rectify (i.e. find the length of) the semi-cubical parabola x3 = ay2, which had been discovered in 1657 by his pupil William Neil. Since all attempts to rectify the ellipse and hyperbola had been (necessarily) ineffectual, it had been supposed that no curves could be rectified, as indeed Descartes had definitely asserted to be the case. The logarithmic spiral had been rectified by Torricelli, and was the first curved line (other than the circle) whose length was determined, but the extension by Neil and Wallis to an algebraic curve was novel. The cycloid was the next curve rectified; this was done by Wren in 1658. Events May 25 - Richard Cromwell resigns as Lord Protector of England following the restoration of the Long Parliament, beginning a second brief period of the republican government called the Commonwealth. ... Blaise Pascal (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... Events January 8 - Miles Sindercombe, would-be-assassin of Oliver Cromwell, and his group are captured in London February - Admiral Robert Blake defeats the Spanish West Indian Fleet in a battle over the seizure of Jamaica. ... Logarithmic spiral (pitch 10°) Cutaway of a nautilus shell showing the chambers arranged in an approximately logarithmic spiral A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. ... Torricelli can refer to the following: Evangelista Torricelli- Italian physicist Robert Torricelli- former American politician Torricelli languages is a subgroup of the Papuan languages This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Genera Donacobius Campylorhynchus Odontorchilus Salpinctes Catherpes Hylorchilus Cinnycerthia Thryomanes Ferminia Troglodytes Cistothorus Uropsila Thryorchilus Henicorhina Microcerculus Cyphorhinus The true wrens are members of a New World passerine bird family Troglodytidae containing 55 species. ... Events January 13 - Edward Sexby, who has plotted against Oliver Cromwell, dies in Tower of London February 6 - Swedish troops of Charles X Gustav of Sweden cross from Sweden to Denmark over frozen sea May 1 - Publication of Hydriotaphia, Urn Burial and The Garden of Cyrus by Thomas Browne September...


Early in 1658 a similar discovery, independent of that of Neil, was made by van Heuraët, and this was published by van Schooten in his edition of Descartes's Geometria in 1659. Van Heuraët's method is as follows. He supposes the curve to be referred to rectangular axes; if this be so, and if (x, y) be the coordinates of any point on it, and n be the length of the normal, and if another point whose coordinates are (x, ) be taken such that : h = n : y, where h is a constant; then, if ds be the element of the length of the required curve, we have by similar triangles ds : dx = n : y. Therefore h ds = dx. Hence, if the area of the locus of the point (x,) can be found, the first curve can be rectified. In this way van Heuraët effected the rectification of the curve y³ = ax² but added that the rectification of the parabola y² = ax is impossible since it requires the quadrature of the hyperbola. The solutions given by Neil and Wallis are somewhat similar to that given by van Heuraët, though no general rule is enunciated, and the analysis is clumsy. A third method was suggested by Fermat in 1660, but it is inelegant and laborious.


The theory of the collision of bodies was propounded by the Royal Society in 1668 for the consideration of mathematicians. Wallis, Wren, and Huygens sent correct and similar solutions, all depending on what is now called the conservation of momentum; but, while Wren and Huygens confined their theory to perfectly elastic bodies, Wallis considered also imperfectly elastic bodies. This was followed in 1669 by a work on statics (centres of gravity), and in 1670 by one on dynamics: these provide a convenient synopsis of what was then known on the subject. 1670 was a common year beginning on a Saturday in countries using the Julian calendar and a Wednesday in countries using the Gregorian calendar. ...


In 1685 Wallis published Algebra, preceded by a historical account of the development of the subject, which contains a great deal of valuable information. The second edition, issued in 1693 and forming the second volume of his Opera, was considerably enlarged. This algebra is noteworthy as containing the first systematic use of formulae. A given magnitude is here represented by the numerical ratio which it bears to the unit of the same kind of magnitude: thus, when Wallis wants to compare two lengths he regards each as containing so many units of length. This perhaps will be made clearer by noting that the relation between the space described in any time by a particle moving with a uniform velocity is denoted by Wallis by the formula s = vt, where s is the number representing the ratio of the space described to the unit of length; while the previous writers would have denoted the same relation by stating what is equivalent to the proposition . It is curious to note that Wallis rejected as absurd the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity. Events January 11 - Eruption of Mt. ...


References

The initial text of this article was taken from the public domain resource A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908). The public domain comprises the body of all creative works and other knowledge—writing, artwork, music, science, inventions, and others—in which no person or organization has any proprietary interest. ...

  • Biography (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Wallis.html) at the MacTutor archive
  • C J Scriba, The autobiography of John Wallis, F.R.S.. Notes and Records Roy. Soc. London 25 (1970), 17-46.

Another John Wallis was also a Conservative Member of Provincial Parliament in the Ontario legislature in the 1st Parliament of Ontario representing Toronto West (September 03, 1867-February 25, 1871). The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... The Ontario Progressive Conservative Party (PC Party of Ontario) is a right-of-centre political party in Ontario, Canada. ... For the U.S. cannabis reform organization, see Marijuana Policy Project. ... The Ontario Legislature Building at Queens Park The Legislative Assembly of Ontario, is the legislature of the Canadian province of Ontario. ... The 1st Parliament of Ontario was in session from September 3, 1867 until February 25, 1871, just prior to the 1871 general election. ... Toronto West was a former federal electoral district represented in the Canadian House of Commons, and located in the province of Ontario. ...


 
 

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